Graphing - GMAT Quantitative

Set in the equation:

The -intercept is .

Set in the equation and solve for .

This is impossible, so the equation has no solution. Therefore, the graph has no -intercept.

The vertical asymptote is , where is found by setting the denominator equal to 0 and solving for :

This is the equation of the vertical asymptote.

Set in the equation and solve for .

The -intercept is

To find the horizontal asymptote, we can divide both numerator and denominator in the right expression by :

As approaches positive or negative infinity, and both approach 0. Therefore, approaches , making the horizontal asymptote the line of the equation .

Set in the equation:

The -intercept is .

Set in the equation and solve for .

This is impossible, so the equation has no solution. Therefore, the graph has no -intercept.

The vertical asymptote is , where is found by setting the denominator equal to 0 and solving for :

This is the equation of the vertical asymptote.

Set in the equation and solve for .

The -intercept is

To find the horizontal asymptote, we can divide both numerator and denominator in the right expression by :

As approaches positive or negative infinity, and both approach 0. Therefore, approaches , making the horizontal asymptote the line of the equation .

The -coordinate(s) of the -intercept(s) are the real solution(s) to the equation . We can use the quadratic formula to find any solutions, setting - the coefficients of the expression.

An examination of the discriminant , however, proves this unnecessary.

The discriminant being negative, there are no real solutions, so the parabola has no -intercepts.

When plotting a complex number, we use a set of real-imaginary axes in which the x-axis is represented by the real component of the complex number, and the y-axis is represented by the imaginary component of the complex number. The real component is and the imaginary component is , so this is the equivalent of plotting the point on a set of Cartesian axes. Plotting the complex number on a set of real-imaginary axes, we move to the left in the x-direction and up in the y-direction, which puts us in the second quadrant, or in terms of Roman numerals:

If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:

We are essentially doing the same as plotting the point on a set of Cartesian axes. We move units right in the x direction, and units down in the y direction, which puts us in the fourth quadrant, or in terms of Roman numerals:

If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:

We are essentially doing the same as plotting the point on a set of Cartesian axes. We move units left of the origin in the x direction, and units down from the origin in the y direction, which puts us in the third quadrant, or in terms of Roman numerals:

If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:

We are essentially doing the same as plotting the point on a set of Cartesian axes. We move units right of the origin in the x direction, and units up from the origin in the y direction, which puts us in the first quadrant, or in terms of Roman numerals:

Squaring an expression, then squaring the result, amounts to taking the original expression to the fourth power. Therefore, we can first square :

Now square this result:

For any expression , . That is, we can raise an expression to the power of eight by squaring it, then squaring the result, then squaring that result.

First, we square:

Square this result to obtain the fourth power:

Square this result to obtain the eighth power:

We first need to find an equation of the line with a slope of that passes through the point (2, 5).

Now, plug in the point and solve for .

The slope of the line through and can be found using the slope formula:

Set :

We can use this slope and the slope formula; set :

In order to find the value of , we first need to find the equation for the line with slope that passes through the point .

Plugging in and solving for :